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A190281
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Decimal expansion of (1+sqrt(1+r))/r, where r=sqrt(2).
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3
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1, 8, 0, 5, 7, 9, 0, 8, 9, 4, 6, 5, 4, 3, 5, 7, 4, 9, 0, 4, 4, 0, 6, 4, 5, 5, 5, 7, 3, 4, 5, 5, 2, 7, 4, 1, 7, 8, 2, 9, 2, 2, 9, 0, 5, 8, 6, 1, 5, 6, 3, 1, 7, 8, 0, 3, 3, 2, 7, 5, 1, 4, 4, 7, 8, 3, 8, 2, 4, 1, 2, 9, 2, 7, 8, 6, 3, 3, 8, 3, 3, 0, 5, 6, 1, 7, 2, 9, 8, 3, 3, 5, 2, 0, 2, 3, 6, 7, 1, 1, 8, 6, 6, 4, 1, 2, 8, 4, 3, 8, 9, 2, 1, 9, 0, 2, 6, 9, 9, 1
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OFFSET
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1,2
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COMMENTS
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The rectangle R whose shape (i.e., length/width) is (1+sqrt(1+r))/r, where r=sqrt(2), can be partitioned into rectangles of shapes sqrt(2) and 2 in a manner that matches the periodic continued fraction [r, 2, r, 2, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1,1,4,6,1,2,2,2,1,1 ...] at A190282. For details, see A188635.
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LINKS
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EXAMPLE
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1.805790894654357490440645557345527417829...
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MATHEMATICA
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r=2^(1/2)
FromContinuedFraction[{r, 2, {r, 2}}]
FullSimplify[%]
ContinuedFraction[%, 100] (* A190282 *)
RealDigits[N[%%, 120]] (* A190281 *)
N[%%%, 40]
RealDigits[(1 + Sqrt[1 + Sqrt[2]])/Sqrt[2], 10, 100][[1]] (* G. C. Greubel, Jan 31 2018 *)
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PROG
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(PARI) (1 + sqrt(1 + sqrt(2)))/sqrt(2) \\ G. C. Greubel, Jan 31 2018
(Magma) (1 + Sqrt(1 + Sqrt(2)))/Sqrt(2); // G. C. Greubel, Jan 31 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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